226 M. Nardon and P. Pianca
proposed for the pricing of both European and American options on dividend paying
stocks, which suggest various model adjustments (such as, for example, subtracting
the present value of the dividend from the asset spot price). Nevertheless, all such
approximations have some drawbacks and are not so efficient (see e.g., Haug [11] for
a review).
Haug and Haug [12] and Beneder and Vorst [2] propose a volatility adjustment
which takes into account the timing of the dividend. The idea behind the approximation
is to leave volatility unchanged before the dividend payment and to apply the adjusted
volatility after the dividend payment. This method performs particularly poorly in the
presence of multiple dividends. A more sophisticated volatility adjustment to be used
in combination with the escrowed dividend model is proposed by Bos et al. [4]. The
method is quite accurate for most cases. Nevertheless, for very large dividends, or in
the case of multiple dividends, the method can yield significant mispricing. A slightly
different implementation (see Bos and Vandermark [5]) adjusts both the stock price
and the strike. The dividends are divided into two parts, called “near” and “far”,
which are used for the adjustments to the spot and the strike price respectively. This
approach seems to work better than the approximation mentioned above. Haug et
al. [13] derive an integral representation formula that can be considered the exact
solution to problems of evaluating both European and American call options and
European put options. Recently, de Matos et al. [7] derived arbitrarily accurate lower
and upper bounds for the value of European options on a stock paying a discrete
dividend.
For American-style put options, it can be optimal to exercise at any time prior to
expiration, even in the absence of dividends. Unfortunately, no analytical solutions
for both the option price and the exercise strategy are available, hence one is generally
forced to numerical solutions, such as binomial approaches. As is well known (see
Merton [14]), in the absence of dividends, it is never optimal to exercise an American
call before maturity. If a cash dividend payment is expected during the lifetime of
the option, it might be optimal to exercise an American call option right before the
ex-dividend date, while for an American put it may be optimal to exercise at any point
in time until maturity.
Lattice methods are commonly used for the pricing of both European and Ameri-
can options. In the binomial model (see Cox et al. [6]), the pricing problem is solved
by backward induction along the tree. In particular, for American options, ateach
node of the lattice one has to compare the early exercise value with the continuation
value.
In this contribution, we analyse binomial algorithms for the evaluation of op-
tions written on stocks which pay discrete dividends of both European and American
types. In particular, we consider non-recombining binomial trees, hybrid binomial
algorithms for both European and American call options, based on the Black-Scholes
formula for the evaluation of the option after the ex-dividend date and up to maturity; a
binomial method which implements the efficient continuous approximation proposed
in [5]; and we propose a binomial method based on an interpolation idea given by
Vellekoop and Nieuwenhuis [17], in which the recombining feature is maintained.
The model based on the interpolation procedure is also extended to the case of multi-